Theme Essential Question: How are functions used to describe relationships that happen in our lives?
Essential Questions:
How do you derive the slope-intercept form of a linear equation?
How do you use the slope-intercept form of a linear equation?
Standards Understand the connections between proportional relationships, lines, and linear equations.
8. EE.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8. F.3.Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Objectives
The student will investigate the attributes of slope(m) and y-intercepts(b) for the slope intercepts of a linear function, y=mx+b.
The student will illustrate/explain the derivation of the equations y = mx (a line through the origin) and y = mx + b (a line with a y-intercept of b).
The student will use the slope-intercept form to graph a line.
The student will use the slope and y-intercept to write the equation in slope-intercept form.
By using coordinate grids and various sets of three similar triangles, students can prove that the slopes of the corresponding sides are equal, thus making the unit rate of change equal. After proving with multiple sets of triangles, students can be led to generalize the slope to y = mx for a line through the origin and y = mx + b for a line through the vertical axis at b.
Assessment Product
Algebra Activity (Glencoe Algebra I, p. 271): Investigating Slope/Intercept Form.
Experiment by collecting data and adding a constant weight to see how it affects the distance of a rubber band.
In our construction theme, this is a simulation of a physical application used to study properties of materials. Complete the experiment and place the write-up in your portfolio as Entry #4.
Key Questions
Where in the coordinate plane does the point containing the y-intercept lie?
What is the x-coordinate of the point that contains the y-intercept?
Given an equation in slope-intercept form, what is the procedure to graph the line on the coordinate plane?
Given the slope and y-intercept, how you would write the equation of the line?
Observable Student Behaviors
The student can provide a viable argument to support the derivation of the equation y = mx for a line through the origin.
The student can provide a viable argument to support the derivation of the equation y = mx + b for a line intercepting the vertical axis at b.
The student can write linear equations in slope-intercept form, given the slope and y-intercept.
The student can graph linear equations from slope-intercept form.
The student can model real world data with an equation in slope-intercept form.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary Input, output, domain, range, independent and dependent variables, linear equation, linear function, rate of change, unit rate, constant rate of change or constant, slope, slope-intercept form, ordered pairs, coordinate plane, proportional relationship, derive
Suggested Activities [see Legend to highlight MCO and HYS]
Houghton Mifflin OnCore Mathematics Middle School Grade 8
Unit 2-4 p.47-48
ABC Mastering Common Core in Mathematics Chapter 5-7 and 5-8, p. 55-56
Gizmo Lessons
Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. (Student Exploration Sheet & Teacher Guide Available)
Slope-Intercept Form of a line-Activity A
Compare the slope‑intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.
Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)
Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
8.EE.6 Activity 1 p. 171, Activity 2 p. 172
8.F.3 p. 189
JBHM Grade 8 GP2, Unit 4, p. 297-328
Glencoe Pre Algebra Chapter 8, Lesson 7, p. 404-408
Glencoe Algebra I Chapter 5, Lesson 3, p. 272-277
Graphing Calculator Investigation, p 278-279
Highly Recommended:
Nothing Available at this Time (EE.6)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
Grade: 8 Unit: 3 Week: 4 Dates: 12/17-12/21
Content: Slope-Intercept Form
Theme Essential Question:
How are functions used to describe relationships that happen in our lives?
Essential Questions:
Standards
Understand the connections between proportional relationships, lines, and linear equations.
Objectives
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials.
**http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf**
(Taken from **Ohio Dept of Education Mathematics Model Curriculum 6-28-2011** )
By using coordinate grids and various sets of three similar triangles, students can prove that the slopes of the corresponding sides are equal, thus making the unit rate of change equal. After proving with multiple sets of triangles, students can be led to generalize the slope to y = mx for a line through the origin and y = mx + b for a line through the vertical axis at b.
Assessment
Product
Key Questions
Observable Student Behaviors
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Vocabulary
Input, output, domain, range, independent and dependent variables, linear equation, linear function, rate of change, unit rate, constant rate of change or constant, slope, slope-intercept form, ordered pairs, coordinate plane, proportional relationship, derive
Suggested Activities [see Legend to highlight MCO and HYS]
- Houghton Mifflin OnCore Mathematics Middle School Grade 8
- Unit 2-4 p.47-48
- ABC Mastering Common Core in Mathematics Chapter 5-7 and 5-8, p. 55-56
- Gizmo Lessons
- Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. (Student Exploration Sheet & Teacher Guide Available)- Slope-Intercept Form of a line-Activity A
Compare the slope‑intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.- Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. (Student Exploration Sheet & Teacher Guide Available)- Teaching the Common Core Math Standards with Hands-On Activities by Judith Muschla
- 8.EE.6 Activity 1 p. 171, Activity 2 p. 172
- 8.F.3 p. 189
- JBHM Grade 8 GP2, Unit 4, p. 297-328
- Glencoe Pre Algebra Chapter 8, Lesson 7, p. 404-408
- Glencoe Algebra I Chapter 5, Lesson 3, p. 272-277
- Graphing Calculator Investigation, p 278-279
Highly Recommended:Nothing Available at this Time (EE.6)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
Diverse Learners
Odyssey (teacher discretion)
Skills Tutor (teacher discretion)
Algebra’scool: Unit C Module 9
Homework
Suggested:
**http://www.kutasoftware.com/free.html** to print assignments on a variety of topics
See appropriate Glencoe, OnCore, JBHM, and ABC materials
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